FFlavor Constraints on New Physics
Zoltan Ligeti ∗ Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USAE-mail: [email protected]
This talk highlights, from a theoretical point of view, some recent exciting results in flavor physics,as well as future prospects. We discuss possible implications of a subset of the experimentalresults in tension with the standard model, such as the 4 σ deviation in the B → D ( ∗ ) τ ¯ ν decayrates, and recent improvements in the constraints on axion portal dark matter models. We use theexamples of constraining new physics contributions to neutral meson mixing and the search forpossible vector-like fermions to illustrate the expected progress over the next decade to increasethe sensitivity to new physics at shorter distance scales. We also speculate about the ultimatelimitations of (quark) flavor physics probes of new physics. XXVII International Symposium on Lepton Photon Interactions at High Energies17-22 August 2015Ljubljana, Slovenia ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - ph ] J un lavor Constraints on New Physics Zoltan Ligeti
1. Introduction
I was asked to talk about flavor physics constraints on new physics (NP). A slight complicationis that in the absence of unambiguous observations of deviations from the standard model (SM) inlaboratory experiments so far, and with the LHC pushing the scale of NP higher, there are not reallygood “simplified models" for non-SM flavor physics, containing a modest number of parameters,which a large class of NP models match onto. There are many flavor physics constraints, andthere are many NP models, and attempts to simplify this large (and for most people uninspiring)matrix of constraints has only achieved limited success for a handful of processes. In other words,the interesting information from flavor physics is not simple (as it depends on a large number ofprocesses and the theory is often complicated), and the simple information from flavor physics isnot interesting; for example, we learn little from just the values of Cabibbo-Kobayashi-Maskawa(CKM) matrix elements .Flavor physics is the study of interactions that distinguish the three generations of fermions,i.e., interactions that break the global [ U ( )] symmetry of the SM, of which each U ( ) acts on oneof the 5 fermion representations ( Q , u d , L , e ). In the SM, this symmetry is broken by the Yukawacouplings, while in the presence of new physics there generally are additional sources of flavor (and CP ) violation. We do not understand the flavor structure of the SM, and if there is new physics atthe 1–100 TeV scale, we have to understand the mechanism that suppresses its effects in the flavorsector to satisfy the experimental constraints. In addition, the observed baryon asymmetry of theUniverse requires CP violation beyond the SM (although that need not occur in the quark sector,nor necessarily in flavor changing processes). In any case, flavor measurements provide rich andsensitive ways to probe the SM and search for NP, and flavor physics strongly constrains any NPwithin the LHC reach. These measurements will reach much higher sensitivities in the next decade.The sensitivity of flavor measurements to very high mass scales typically comes from largeSM suppressions. As a simple (and historically important) example, consider K – K mixing.The splitting between the two mass eigenstates is ∆ m K / m K ∼ × − . In the SM, ∆ m K arisesdominantly from box diagrams with virtual W bosons and c quarks, and can be estimated as ∆ m K m K ∼ α w | V cs V cd | m c m W f K . (1.1)The result is suppressed by CKM angles, a loop factor, the weak coupling, and the GIM mechanism.If a heavy particle, X , with effective ¯ sdX coupling, g , contributes an O ( ) fraction to ∆ m K , then (cid:12)(cid:12)(cid:12)(cid:12) ∆ m ( X ) K ∆ m ( exp ) K (cid:12)(cid:12)(cid:12)(cid:12) ∼ (cid:12)(cid:12)(cid:12)(cid:12) g Λ M X ∆ m ( exp ) K (cid:12)(cid:12)(cid:12)(cid:12) ⇒ M X g (cid:38) × TeV . (1.2)So even TeV-scale particles with loop-suppressed couplings [ g ∼ O ( − )] can give observableeffects. Thus, flavor measurements probe the TeV scale if the NP has SM-like flavor structure,and much higher scales if the NP flavor structure is generic. In the SM only W bosons changefermion flavor, so flavor-changing processes of the known fermions (except the t quark) are sup-pressed by at least the second power of a high scale ( G F ∝ / m W in the SM, or 1 / M ) and often As Lincoln Wolfenstein sometimes said, even though he invented the Wolfenstein parameters, he did not care whattheir values were, only whether many overconstraining measurements gave consistent determinations. lavor Constraints on New Physics Zoltan Ligeti γγα α d m ∆ K ε K ε s m ∆ & d m ∆ ub V β sin 2 (excl. at CL > 0.95) < 0 β sol. w/ cos 2 e xc l uded a t C L > . α βγ ρ η excluded area has CL > 0.95EPS 15 CKM f i t t e r
Figure 1:
The standard model CKM fit, and individual constraints (colored regions show 95% CL) [5]. by additional small coefficients. We want to find out if the higher dimension operators generatedby the high-scale physics have coefficients as predicted by the SM, and if operators forbidden inthe SM (e.g., right handed currents) are generated.While new physics has been widely expected to occur at the TeV scale, hinted by the hierarchyproblem and the WIMP paradigm, after the discovery of the Higgs boson, no other new particleis guaranteed to be observable in near future laboratory experiments. In flavor physics, typicallykaons probe the highest scales, since the SM suppressions are strongest for flavor-changing neutralcurrents (FCNC) between 1st and 2nd generation quarks. In many NP scenarios the 3rd generationis rather different from the first two, so there is strong motivation to explore what the technologyallows us to probe. I find it fairly certain that if new physics is discovered, we will eventuallyunderstand it as “natural", no matter how “strange" it might seem at first.Section 2 summarizes the current status of (quark) flavor physics and reviews some tensionswith the SM predictions. These are some of the most often discussed topics recently, and they arealso interesting because they may have the best chance to be established as clear deviations from theSM, as more data is accumulated. (I include four recent measurements [1, 2, 3, 4] for completeness,which appeared since the conference.) Section 3 gives some examples of the expected futureprogress and improvements in sensitivity to NP, independent of the current data. Section 4 containssome comments on the ultimate sensitivity of flavor physics experiments to NP.
2. Status of flavor physics
A detailed introduction to flavor physics is omitted here, as well as a review of the determi-nations of CKM elements; see, e.g., Refs. [6, 7]. The magnitudes of CKM elements are mainlyextracted from semileptonic and leptonic K , D , and B decays, and B d , s mixing. These determinethe sides of the unitarity triangle shown in Fig. 1, which is a convenient way to compare many3 lavor Constraints on New Physics Zoltan Ligeti
Figure 2:
Some recent measurements in tension with the SM. The horizontal axis shows the nominal sig-nificance. The vertical axis shows (monotonically, in my opinion) an undefined function of an ill-definedvariable: the theoretical cleanliness. That is, the level of plausibility that a really conservative estimate ofthe theory uncertainty of each observable may affect the significance of its deviation from the SM by 1 σ . constraints on the SM and visualize the level of consistency. Any constraint which renders the areaof the unitarity triangle nonzero, such as nonzero angles (mod π ), has to measure CP violation, andwere reviewed in another talk [8]. Some of the most important measurements are shown in Fig. 1,together with the CKM fit in the SM. (The notation ¯ ρ , ¯ η instead of ρ , η simply corresponds toa small modification of the original Wolfenstein parametrization, to keep unitarity exact.) WhileFig. 1 shows very good consistency, it does not address how large new physics contributions areallowed. As we see below, in the presence of new physics the fit becomes less constrained, and O ( ) NP contributions to most FCNC processes, relative to the SM, are still allowed.Several measurements show intriguing deviations from the SM predictions. Some of those thatreach the 2 − σ level are depicted schematically in Fig. 2. The horizontal axis shows the nominalsignificance and the vertical axis relates to the theoretical cleanliness of the SM predictions. WhatI mean is some (monotonic) measure of the plausibility that a conservative estimate of the theoryuncertainty may affect the overall significance by 1 σ . All of these are frequently discussed, somehave triggered hundreds of papers, and could be the subjects of entire talks each.Currently, the B → D ( ∗ ) τ ¯ ν rates, specifically the R ( D ( ∗ ) ) = Γ ( B → D ( ∗ ) τ ¯ ν ) / Γ ( B → D ( ∗ ) l ¯ ν ) ratios (where l = e , µ ) constitute the most significant discrepancy from the SM in collider experi-ments [9, 10, 11, 12, 1] (aside from neutrino masses). The effect is at the 4 σ level [13]. Figure 3shows the current data, the SM expectations, as well as the expected Belle II sensitivity. Thesemeasurements show good consistency with one another. The theory is also on solid footing, sinceheavy quark symmetry suppresses model independently the hadronic physics needed for the SMprediction, most of which is actually constrained by the measured B → D ( ∗ ) l ¯ ν decay distributions.4 lavor Constraints on New Physics Zoltan Ligeti R ( D ) R ( D ∗ ) BaBar 0 . ± . ± .
042 0 . ± . ± . . ± . ± .
026 0 . ± . ± . . ± . ± . . ± . ± . . ± . ± .
028 0 . ± . ± . . ± .
010 0 . ± . ± . ± . R(D) R ( D * ) BaBar, PRL109,101802(2012)Belle, PRD92,072014(2015)LHCb, PRL115,111803(2015)Belle, arXiv:1603.06711) = 67% c HFAG Average, P(SM prediction = 1.0 cD R(D), PRD92,054510(2015)R(D*), PRD85,094025(2012)
HFAG
Prel. Winter 2016
Figure 3:
Left: measurements of R ( D ( ∗ ) ) [9, 11, 12, 1], their averages [13], the SM predictions [14, 15, 16],and future sensitivity [17]. Right: the measurements, world average (red), and SM prediction (magenta). It is somewhat surprising to find such large deviations from the SM in processes which occur attree level in the SM. The central values of the current world averages would imply that there has tobe new physics at a fairly low scale. Some scenarios are excluded by LHC Run 1 bounds already,and many more will soon be constrained by LHC Run 2 data. To fit the current central values,mediators with leptoquark or W (cid:48) quantum numbers are preferred compared to scalars. Leptoquarksare favored if one requires the NP to be minimally flavor violating (MFV), which helps explainthe absence of other flavor signals and suppress direct production of the new particles at the LHCfrom partons abundant in protons [18]. There are several options for the lepton-flavor structureof the new physics, which can have “lepton-MFV" or “ τ -alignment" [18]. For example, the lattercan be realized with an A -type symmetry for the leptons, which links it to neutrino flavor [19].To illustrate the wide range of possibilities, there are viable scenarios in which B → D ( ∗ ) τ ¯ ν areSM-like, but B → D ( ∗ ) l ¯ ν are suppressed by interference between NP and the SM [20].There are many further experimental measurements that can be done to clarify this anomaly.The B → D ( ∗ ) τ ¯ ν rates seem to exceed [18] the LEP measurements of the inclusive b → X τ ¯ ν rate [7],and the inclusive B → X c τν rate [21] has not yet been measured. The equality of the e and µ rates are not well constrained, and the currently allowed differences [22, 23] open up (or keepopen) model building options [24]. In many scenarios, bounds on b → s ν ¯ ν processes are veryimportant [18, 25]. A lot will be learned, hopefully soon, from LHCb result on R ( D ) , measurementsusing hadronic τ decays, measurements in Λ b and B s decays, and later from Belle II. If a deviationfrom the SM is established, it will strongly motivate to measure all possible semitauonic modes,both in b → c and b → u transitions [26, 27].Another measurement which has drawn immense attention is the “ P (cid:48) anomaly" in a B → K ∗ µ + µ − angular distribution (see, e.g., Refs. [28, 29]), measured at LHCb [30] and recently atBelle [2], and discussed in another talk in more detail [31]. The measurements are shown in theleft plot in Fig. 4, together with a SM prediction [32]. These “optimized observables" are based onthe SCET factorization theorem for semileptonic B decay form factors [33, 34], and constructingcombinations from which the “nonfactorizable" (“soft") contributions cancel. (These are nonper-turbative functions of q , which obey symmetry relations [35]; additional terms are either powersuppressed or contain an explicit α s factor.) The magnitudes of the correction terms, that is one’sability to calculate the form factor ratios at small q reliably, is debated [36] (and are not well con-strained by data yet). The tension between theory and the data is certainly intriguing, and many5 lavor Constraints on New Physics Zoltan Ligeti q (GeV /c ) P Belle preliminary
This AnalysisLHCb 2013LHCb 2015SM from DHMV ] − [10) − µ + µ → s0 B B(0 1 2 3 4 5 6 7 ] − [ ) − µ + µ → B B ( − . % . % . % ATLAS = 7 TeV, 4.9 fbs = 8 TeV, 20 fbsATLAS SM ) = 2.3, L ln( ∆ Contours for 2 L Figure 4:
Left: The LHCb [30] and Belle [2] measurements of P (cid:48) in B → K ∗ µ + µ − . Right: The B s , d → µ + µ − measurements from LHCb and CMS [44], and the ATLAS constraints [3] superimposed. studies exist both in terms of model independent fits and specific model predictions. Some of thesimplest models are Z (cid:48) -like, with nonuniversal flavor couplings. One may be concerned that thebest fit is a new contribution to the operator O = e ( ¯ s γ µ P L b )( ¯ (cid:96) γ µ (cid:96) ) in the effective Hamiltonian,the same term which would be modified if theoretical control over the c ¯ c loop contributions wereworse than expected. (This was also emphasized recently in Ref. [37].) There are many possibleconnections to the ∼ . σ anomaly in Γ ( B → Ke + e − ) (cid:54) = Γ ( B → K µ + µ − ) as well [38].For these observables, too, I trust that with improved measurements and theory, the source ofthe currently seen effects will be understood. With more data, one can test the q (in)dependenceof the extracted Wilson coefficients. In the large q (small recoil) region one can make modelindependent predictions both for exclusive [39] inclusive [40] b → sl + l − mediated decays, whichis complementary to the small q region, and has different theory uncertainties.Another anomaly observed is the B s → φ µ + µ − rate in the 1 < q < region being about3 σ below theoretical calculations [41]. This relies on QCD sum rules [42] combined with latticeQCD calculations of the form factors at large q [43]. Extending the lattice results to lower q would help clarify the picture, as well as more precise measurements, also at high q .If new physics is at play in these processes, it is likely to impact B → µ + µ − , too. The com-bined LHCb and CMS observation [44] of B s → µ + µ − , the constraint on B d → µ + µ − , and therecent ATLAS [3] constraints are shown in the right plot in Fig. 4, as well as the SM prediction.Measuring a rate at the 3 × − level is impressive, and future refinements are high priority. Thenonperturbative input in this case is just f B , which is under good control in lattice QCD.Another deviation from the SM expectations, which is theoretically very clean, and has been3 − σ , is the DØ measurement of the like-sign dimuon charge asymmetry in semileptonic decaysof b hadrons, ( N µ + µ + − N µ − µ − ) / ( N µ + µ + + N µ − µ − ) [45], shown in the left plot in Fig. 5. A nonzerosignal could come from a linear combination of CP violation in B s and B d mixing, a d , s SL (see, e.g.,Ref. [46]), and the SM prediction is well below the current sensitivity. Separate measurements of a d SL and a s SL from BaBar, Belle, DØ, and LHCb are consistent in with the SM. The very recentLHCb measurement of a s SL = ( . ± . ) % with 3/fb [4], reducing the uncertainty from 0.62%with 1/fb, starts to be in tension with the DØ anomaly. If there is new physics in CP violation in B s mixing, then one may also expect to see a deviation from the SM in the time-dependent CP asym-6 lavor Constraints on New Physics Zoltan Ligeti [%] d sl a - - - [ % ] s s l a - - - -
01 Standard Model X nm (*) D LHCb X nm (*) D D0 n l * D BaBar ll BaBar ll Belle mm D X nm s D D X nm s D L H C b CMS20 fb CDF 9.6 fb Dfl 8 fb SM
68% CL contours( )
LHCbLHCbLHCbLHCbCombined3 fb Figure 5:
Left: bounds on CP violation in B d , s mixing, a d , s SL [4]. The vertical and horizontal bands show theaverages of the separate B d and B s measurements, respectively, and the yellow ellipse is the DØ measure-ment. Right: measurements of φ s ≡ − β s showing good consistency with the SM. metry in B s → J / ψφ . Recent LHC measurements, however, are consistent with the SM, as shown inthe right plot in Fig. 5. Most importantly, the theory uncertainties are well below the experimentalsensitivity in the coming years, so a lot can be learned from more precise measurements.Understanding the long-standing tensions between inclusive and exclusive measurements of | V cb | and | V ub | are important for NP searches. The ratio | V ub / V cb | together with γ determine the apexof the unitarity triangle from tree-level processes, which is crucial for improving the sensitivity toNP in B mixing and in CP violation measurements involving loop processes. Understanding theQCD dynamics of semileptonic B decays is also important, because the theoretical tools coincidewith those used in inclusive and exclusive b → s γ and b → s (cid:96) ¯ (cid:96) decays. While I have also entertainedmodifications of these CKM measurements due to NP (such as right-handed currents [47]), manyknown theoretical and experimental improvements can take place in the future, such as doing allmeasurements in events where the other B is fully reconstructed.The muon g − σ tension with the SM, according to mostanalysis. It may be a sign of new physics, but some complicated strong interaction dynamics couldstill be at play and decrease the significance of the deviation. One hopes that lattice QCD willdetermine reliably the hadronic light-by-light scattering and vacuum polarization contributions.While supersymmetric models with relatively light sleptons can still account for the deviation [48],the LHC will improve limits on such NP explanations, and sub-GeV weakly coupled new particlescould also be at play [49]. For the heavy NP explanation, somewhat puzzling is that the requirednew physics contribution is the same size as the one-loop SM electroweak contribution.It has long been known that kaon CP violation is sensitive to some of the highest energyscales. For the ε parameter, the SM is in good agreement with the data, and the NP contributionis constrained to be < ∼
30 % of that of the SM [50]. Calculating the SM prediction for direct CP violation, the ε (cid:48) parameter, has been a multi-decade challenge, and important progress has beenmade recently [51]. Results with several lattice spacings are needed to decide if NP is present.These experimental hints of possible deviations from the SM are fantastic for several reasons.Unambiguous evidence for NP would obviously be the start of a new era, and one would also geta rough upper bound on the scale of NP, even if it is not seen directly at ATLAS & CMS. It is also7 lavor Constraints on New Physics Zoltan Ligeti ) [MeV] - m + m ( m ) = T e V h ( m [ P e V ] , b ) t a n c ( f ) = T e V h ( m [ P e V ] , b ) t a n c ( f hadrons) = 0 fic ( B hadrons) = 0.99 fic ( B LHCb Β m H ! G e V " Bound on f a FIG. 1: Bounds on f a as a function of tan β and m H for n = 1in Eq. (8), for m a ≪ m B . For each displayed value of f a thereare two contour lines, and the region between them is allowedfor f a below the shown value. The bound disappears alongthe dashed curve, and gets generically weaker for larger tan β . that LHCb should be able to carry out a precise mea-surement [40]. Interestingly, since the B → Ka signal isessentially a delta function in q , the bound in Eq. (15)can be improved as experimental statistics increase byconsidering smaller and smaller bin sizes, without beinglimited by theoretical uncertainties in form factors [41](or by nonperturbative contributions [42]). The boundon f a will increase compared to the results we obtain inthe next section, simply by scaling with the bound on1 / ! Br( B → Ka ). V. INTERPRETATION
We now derive the bounds on f a using the calculated B → Ka branching ratio in Eq. (14) and the experimen-tal bound in Eq. (15). We start with the axion portalscenario with Br( a → µ + µ − ) ∼ θ isdefined in terms of f a by Eq. (8). We will then look atthe bound on more general scenarios, including the lightHiggs scenario in the NMSSM.For the axion portal, Fig. 1 shows the constraints on f a as a function of the charged Higgs boson mass m H andtan β . For concreteness, we take n = 1; other values of n correspond to a trivial scaling of f a . In the mass rangein Eq. (1), the dependence on m a is negligible for settinga bound. The bound on f a is in the multi-TeV range forlow values of tan β and weakens as tan β increases. Ateach value of tan β , there is a value of m H for which the m H ! GeV " f a t a n Β ! T e V " Bound on f a tan Β ! Large tan Β " FIG. 2: The shaded regions of f a tan β are excluded in thelarge tan β limit. To indicate the region of validity of thelarge tan β approximation, the dashed (dotted) curve showsthe bound for tan β = 3 (tan β = 1). b → sa amplitude in Eq. (12) changes signs, indicated bythe dashed curve in Fig. 1, along which the bound dis-appears. Higher order corrections will affect where thiscancellation takes place, but away from a very narrow re-gion near this dashed curve, the derived bound is robust.The region tan β < β , the X pieceof Eq. (12) dominates, and sin(2 β ) / / tan β + O (1 / tan β ). In this limit, the constraint takes a par-ticularly simple form that only depends on the combi-nation f a tan β , as shown in Fig. 2. Except in the re-gion close to m H ∼
550 GeV, the bound is better than f a tan β > ∼ few ×
10 TeV.These B → Ka bounds are complementary to thoserecently set by BaBar [30] in Υ( nS ) → γ a → γ µ + µ − : f a > ∼ (1 . × sin β . (16)For example, for m H ≃
400 GeV, the Υ bound dominatesfor tan β > ∼
5, while B → Ka dominates for tan β < ∼ m H and tan β are free parameters.One would like some sense of what the expected valuesof m H and tan β might be in a realistic model. Ref. [8]considered a specific scenario based on the PQ-symmetricNMSSM [31]. In that model small tan β is preferred,since large tan β requires fine-tuning the Higgs potential.In addition, m H is no longer a free parameter and isapproximately related to the mass of the lightest CP -even scalar s via m H ≃ m W + " β m s f a v EW . (17) Freytsis, Ligeti, Thaler[0911.5355] LHCb, m(a) = 600 MeV[1508.04094]0140280420560700 m H ± (GeV) Figure 6:
Left: LHCb bounds on f χ tan β as a function of m µ + µ − [57] in the model [59]. Right: The boundas a function of m H ± in the same model; the right axis shows a nearly order of magnitude improvement. useful to have experimental results challenge theory, since unexpected signals motivate both modelbuilding and revisiting the SM predictions. This was the case with the Tevatron anomaly in the t ¯ t forward-backward asymmetry, A t ¯ t FB , which disappeared due to refinements of the experimentalresults (the SM predictions also improved [52]). Concerning the recent 3 σ hint for direct CP violation in the difference of CP asymmetries in D → K + K − and D → π + π − , ∆ A CP = A K + K − − A π + π − , I doubt the initial measurement near 1% could be attributed to the SM [53]. The centralvalue of the world average has decreased since 2012, as has the significance of the hint for ∆ A CP (cid:54) =
0. We probably still do not know how large ∆ A CP the SM could generate. However, exploring ittaught us, for example, about how much (or how little) the quark and squark mixing matrices candiffer and squark masses (don’t) need to be degenerate [54, 55] in alignment models [56].A recent measurement in which no anomaly is seen, but I find the nearly order of magnitudeincrease in mass-scale sensitivity due to a recent LHCb analysis [57] impressive, is the search foran axion-like particle, coupling to SM fermions as ( m ψ / f a ) ¯ ψγ ψ a , explained in another talk [31].Such models are also interesting, because they may have highly suppressed spin-independent directdetection cross sections [58]. The left plot in Fig. 6 shows the 95% CL lower bound on f χ tan β in the model of Ref. [59], from the absence of a narrow µ + µ − peak in B → K ∗ χ ( χ → µ + µ − ) as afunction of m µ + µ − . The bound is shown for m H ± = m H ± ( f a in [59] is f χ in [57]). The left vertical axis is the bound estimated in 2009 [59] fromBaBar & Belle data with only a few bins, and the right vertical axis shows the LHCb bound [57].The dashed (dotted) curve shows the bound for tan β = β = β , the NP contribution vanishes due to a cancellation for a certain value of m H ± .
3. Future increases in new physics scales probed
I would like to talk about three topics briefly in this part: (i) the future theory uncertainty ofthe measurement on sin 2 β from B → J / ψ K S ; (ii) the future sensitivity to NP in mixing of neutralmesons; (iii) sensitivity of flavor physics experiments to very heavy vector-like fermions.8 lavor Constraints on New Physics Zoltan Ligeti sin β ? The theoretical uncertainty of the SM predictions for the time dependent CP asymmetries inthe “gold-plated" modes B → J / ψ K S and B s → J / ψφ are of great importance. They arise fromcontributions to the decay amplitude proportional to V ub V ∗ us instead of the dominant V cb V ∗ cs terms. Irefer to this as V ub contamination, instead of the often used penguin pollution phrase (which is lesscorrect and less clear). This effect did not matter in practice in the past, but it will be importantfor interpreting the full LHCb and Belle II data sets. During the BaBar/Belle era, the experimentalprecision was an order of magnitude above the nominal magnitude of the theoretical uncertainty, λ ( α s / π ) ∼ . A numberof approaches have been developed, using a combination of diagrammatic and flavor symmetryarguments with various assumptions [62, 63]. (I hasten to add a triviality: there is no relation basedonly on SU ( ) flavor symmetry between final states which are entirely in different representations;e.g., φ is an SU ( ) singlet and ρ & K ∗ are members of an octet.) The experimental tests performedso far [8] do not indicate big enhancements of the theory uncertainties.The question that really matters in my opinion is not what it takes to set plausible upper boundson the V ub contamination, while the measurements agree with the SM, but what it would take toconvince the community that NP is observed at LHCb and Belle II, especially if no NP is seen byATLAS and CMS. Therefore, one cannot overemphasize the importance of starting from rigoroustheoretical foundations, with well defined expansion parameter(s).A relation based only on SU ( ) flavor symmetry, which cancels the V ub contamination in sin 2 β against other observables in the SU ( ) limit, was constructed recently [64]sin 2 β = S K S − λ S π − ( ∆ K + λ ∆ π ) tan γ cos 2 β + λ . (3.1)Here S h ( h = K , π ) is the usual coefficient of the sin ( ∆ mt ) term in the time-dependent CP asym-metry [7] in B → J / ψ h , λ (cid:39) .
225 is the Wolfenstein parameter, ∆ h = ¯ Γ ( B d → J / ψ h ) − ¯ Γ ( B + → J / ψ h + ) ¯ Γ ( B d → J / ψ h ) + ¯ Γ ( B + → J / ψ h + ) , (3.2)and ¯ Γ denotes the CP averaged rates. Using Eq. (3.1) it is possible to replace the V ub contamina-tion in the sin 2 β (cid:39) S K S relation with isospin breaking, which could be smaller than the possiblyenhanced V ub contamination one wants to constrain. It also provides redundancy, replacing onetheory uncertainty with a different one. For the V cb V ∗ cs terms in the effective Hamiltonian, ∆ K , π violate isospin, but the V ub V ∗ us terms generate nonzero ∆ h even in the isospin limit. The resultingconstraint on the ¯ ρ − ¯ η plane is shown in Fig. 7 [40].Measuring all terms in Eq. (3.1) is not straightforward. Many of the current measurementsof ∆ h and the production asymmetry of B + B − vs. B B in ϒ ( S ) decay, f + − / f , are circular(the measurements of either assume that the other asymmetry vanishes) [65], so the slight tensionin Fig. 7 should be interpreted with caution. To disentangle ∆ h from the production asymmetry, Until 1997 this estimate was often written as λ ( α s / π ) . Omitting the factor 4 anticipates some enhancement ofthe penguin matrix element, observed in charmless B decays [60] but not yet well constrained in decays to charmonia.Better calculable O ( − ) effects arise from CP violation in K and B mixing, and the Γ B L − Γ B H width difference [61]. lavor Constraints on New Physics Zoltan Ligeti ¯ η ¯ ρ Figure 7:
The dark (light) blue region shows the 1 σ (2 σ ) constraint in the ¯ ρ − ¯ η plane from Eq. (3.1) [64]. more precise measurements of the latter are needed. One option may be to utilize that isospinviolation in inclusive semileptonic decay is suppressed by Λ / m b [65]. (Similar suppression of SU ( ) symmetry breaking in inclusive B decays by Λ / m b is the basis for a theoretically cleanprediction for the ratio Γ ( B → X s (cid:96) + (cid:96) − ) / Γ ( B → X u (cid:96) ¯ ν ) at large q [40].)It is an open question how well it will be possible to ultimately constrain (convincingly) thesize of V ub contamination in the measurements of sin 2 β and 2 β s ( ≡ − φ s ) . Given that SU ( ) flavorsymmetry has been used to analyze B decays for decades, and previously unknown SU ( ) relationscan be discovered in 2015, makes me optimistic that a lot more progress can be achieved. Although the SM CKM fit in Fig. 1 shows impressive and nontrivial consistency, the impli-cations of the level of agreement are often overstated. Allowing new physics contributions, thereare a larger number of parameters related to CP and flavor violation, and the fits become less con-straining. This is shown in Fig. 8, which shows the determination of the unitarity triangle fromtree-dominated decays only, which are unlikely to be affected by new physics. The plot on theleft shows the current fit results, while the constraints in the plot on the right is expected to beachievable with 50 ab − Belle II and 50 fb − LHCb data [66]. The allowed region in the left plot isindeed significantly larger than in Fig. 1.It has been known for decades that the mixing of neutral mesons is particularly sensitive tonew physics, and probe some of the highest scales. In a large class of models, NP has a negligibleimpact on tree-level SM transitions (e.g., the measurements of γ , | V ub | , and | V cb | ), and the 3 × B and B s meson mixing, which can be parametrized as M = M SM12 ( + h q e i σ q ) , q = d , s . (3.3)The constraints on h d and σ d in the B d mixing are shown in Fig. 9, and the constraint in the h d − h s plane is shown in Fig. 10. Both plots show the current constraints (left) and those expected to be10 lavor Constraints on New Physics Zoltan Ligeti γγ ) α ( γ ) α ( γ ub V ub V) & α ( γ & γ α βγ ρ η excluded area has CL > 0.95 2013 CKM f i t t e r ) _ ( a ) _ ( a aa ub V ub V) & _ ( a & a _ ‘a l -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 d -1.5-1.0-0.50.00.51.01.5 excluded area has CL > 0.95 Stage II CKM f i t t e r _ Figure 8:
Constraints on ¯ ρ − ¯ η , allowing NP in the B d , s mixing amplitudes (left) and the expectation using50 ab − Belle II and 50 fb − LHCb data (right) [66]. Colored regions show 95% CL, as in Fig. 1. d h d σ p value excluded area has CL > 0.95 2013 CKM f i t t e r d h d σ p value excluded area has CL > 0.95 Stage II CKM f i t t e r
Figure 9:
Constraints on the h d − σ d parameters (left) and those estimated to be achievable using 50 ab − Belle II and 50 fb − LHCb data (right) [66]. Colored regions show 2 σ limits with the colors indicating CLas shown, while the dashed curves show 3 σ limits. achievable with 50 ab − Belle II and 50 fb − LHCb data (right) [66]. Figure 9 shows that in thefuture the bounds on the “MFV-like regions", where NP flavor is aligned with the SM (2 θ d (cid:39) π ), will be comparable to generic values of the NP phase, unlike in the past. Figure 10 showsthat the bounds on NP in B s mixing, which were significantly weaker than those in the B d sectoruntil recent LHCb measurements, are now comparable, and will comparably improve in the future.As an example, if NP modifies the SM operator describing B q mixing by adding to it a term C q Λ ( ¯ b L γ µ q L ) , (3.4)then one finds h q (cid:39) | C q | | V ∗ tb V tq | (cid:18) . Λ (cid:19) . (3.5)11 lavor Constraints on New Physics Zoltan Ligeti d h s h p value excluded area has CL > 0.95 2013 CKM f i t t e r
HFAG 2014 s φ d h s h p value excluded area has CL > 0.95 Stage II CKM f i t t e r
Figure 10:
Constraints on the h d − h s parameters now (left plot) and those estimated with 50 ab − Belle IIand 50 fb − LHCb data (right plot) [66]. The notation is the same as in Fig. 9.
Couplings NP loop Scales (TeV) probed byorder B d mixing B s mixing | C q | = | V tb V ∗ tq | tree level 17 19(CKM-like) one loop 1.4 1.5 | C q | = × × (no hierarchy) one loop 2 × Table 1:
The scale of the B d , s mixing operators in Eq. (3.4) probed, with 50 ab − Belle II and 50 fb − LHCbdata [66]. The differences due to CKM-like hierarchy of couplings and/or loop suppression is shown.
We can then translate the plotted bounds to the scale of new physics probed. The summary ofexpected sensitivities are shown in Table 1. The sensitivities even with SM-like loop- and CKM-suppressed coefficients are comparable to the scales probed by the LHC in the next decade.In K – K mixing the simplest analog of Eq. (3.3) is to parameterize NP via an additive termto the so-called tt contribution in the SM, M K , tt = M K , tt ( + h K e i σ K ) . The reason is the shortdistance nature of NP and the fact that in many NP models the largest contribution to M K arise viaeffects involving the third generation. Substantial progress would require lattice QCD to constrainthe long distance contribution to M K at the percent level [66].There are also strong constraints on NP from D – D mixing. Since the observed mixing pa-rameters are probably dominated by long distance physics [67], it is hard to improve the bound fromsimply demanding the NP contribution to be below the measured values of the mixing parameters. Another illustration of the expected progress with well quantifiable increases in mass scalesensitivity, in both quark and lepton flavor experiments, is to consider extensions of the SM in-volving vector-like fermions, which can Yukawa couple to the SM [68]. These fermions can havemasses M much greater than the weak scale, since they have a mass term even in the absence of12 lavor Constraints on New Physics Zoltan Ligeti D R H D R H s L d L d L s L D R H + D R H + c L u L u L c L Figure 11:
One-loop vector-like fermion contributions to K and D mixing in Model V [68]. electroweak symmetry breaking. These models are a class of simple extensions of the SM, whichdo not worsen the hierarchy puzzle. There are 11 renormalizable models [68] which add to theSM vector-like fermions in a single (complex) representation of the gauge group that can Yukawacouple to the SM fermions through the Higgs field (4 to leptons, 7 to quarks).The precise definitions of the λ i Yukawa couplings depend on the models, as do the formsof the Lagrangians. For example, what was labeled Model V in Ref. [68] contains vector-likefermions, D , with the same quantum numbers as the SM right-handed down-type quarks, whichYukawa couple to the SM left-handed quark doublets Q iL as L ( V ) NP = ¯ D ( i / D − M ) D − ( λ i ¯ D R H † Q iL + h . c . ) , (3.6)These new interactions generate Z couplings, e.g., in this Model V to the quarks, L ( V ) Z = − ∑ i , j (cid:18) λ ∗ i λ j m Z g Z M (cid:19) ¯ d iL γ µ d jL Z µ , (3.7)which contribute to, and are constrained by, flavor-changing neutral currents.These models also generate dimension-6 four-fermion operators, which contribute to neu-tral meson mixing. At tree level, the Z contribution in Eq. (3.7) yields coefficients of the form ( λ i λ ∗ j ) v / M . At one loop, coefficients of order ( λ i λ ∗ j ) / ( π M ) are generated, which are neitherCKM nor quark-mass suppressed, seemingly not considered in the literature. For large M , theseone-loop contributions are more important than tree-level Z exchange. They are independent ofthe Higgs vacuum expectation value, v , and arise from short distances ∼ / M . They can be calcu-lated in the symmetric phase from the box diagrams in Fig. 11 with virtual scalars and the heavyvector-like fermions. The resulting effective Lagrangian in Model V is [68], L ( V ) mix = − ( λ ∗ i λ j ) π M (cid:20) ∑ klmn (cid:0) ¯ u kL V ki γ µ V † jl u lL (cid:1)(cid:0) ¯ u mL V mi γ µ V † jn u nL (cid:1) + (cid:0) ¯ d iL γ µ d jL (cid:1)(cid:0) ¯ d iL γ µ d jL (cid:1)(cid:21) + h . c . (3.8)Table 2 summarizes the bounds on 5 of the 11 models for illustration. The upper rows foreach model show the current bounds, and the lower rows show the expected sensitivities in thenext generation of experiments (in the next decade or so). For the vector-like fermions that coupleto SM quarks, the bounds are shown separately from ∆ F = ∆ F = ∆ F = ε K probes much higher scales than ∆ m K in these models. (In the other casesthe differences are of order unity.) We learn that the next generation of experiments will improvethe mass scale sensitivities in the leptonic (hadronic) models by up to a factor of ∼ ∼ lavor Constraints on New Physics Zoltan Ligeti
Model Quantum Bounds on M / TeV and λ i λ j for each i j pairnumbers i j = i j = i j = ( , , − ) a b c a b c III ( , , − / ) a b c a b c ∆ F = ∆ F = ∆ F = ∆ F = ∆ F = ∆ F = ( , , − / ) d [100] e {42, 670} f g h i j d {100, 1000} f l h k j VII ( , , − / ) d [71] e {47, 750} f g h i j d {110, 1100} f l h k j XI ( , , − / ) d [100] e {42, 670} f g h k j d {100, 1000} f l h k j Table 2:
Bounds in some of the vector-like fermion models [68] on M [ TeV ] / (cid:112) | λ i λ j | in the leptonic models,and from the ∆ F = ∆ F = M / (cid:112) | λ i λ j | , exceptfor K meson mixing we show (cid:8) M / (cid:113) | Re ( λ i λ ∗ j ) | , M / (cid:113) | Im ( λ i λ ∗ j ) | (cid:9) . The strongest bounds arise, or areexpected to arise, from: a ) µ to e conversion; b ) τ → e π ; c ) τ → µρ ; d ) K → πν ¯ ν ; e ) K L → µ + µ − (thisinvolves | Re ( λ λ ∗ ) | and is in square brackets because prospects for improvements are weak); f ) K mixing; g ) B → π µ + µ − ; h ) B d mixing; i ) B → X s (cid:96) + (cid:96) − ; j ) B s mixing; k ) B s → µ + µ − , l ) B d → µ + µ − . These are vast topics which I could not cover in detail in the talk, nor is it possible here.Top quarks in the SM decay almost exclusively to bW , with the second largest branchingfraction B ( t → sW ) < × − . Particularly clean probes of the SM are FCNC top decays, forwhich the SM predictions are below the 10 − level. The current bounds are roughly at the level B ( t → qZ ) < ∼ − , B ( t → qg ) < ∼ − , and B ( t → qh ) < ∼ . q = u , c produced by new physics [69]. The ultimate LHC sensitivities are expectedto be about a factor of 10 better, hence any observation would be a clear sign of NP. There isobvious complementarity between FCNC searches in the top sector, and low energy flavor physicsbounds. Since t L is in the same SU ( ) doublet as b L , several operators have correlated effects in t and b decays. For some operators, mainly those involving left-handed quark fields, the low energyconstraints exclude a detectable LHC signal, whereas other operators are still allowed to have largeenough coefficients to yield detectable NP signals at the LHC (see, e.g., Ref. [70]).The experimental richness of higgs physics, that several production mechanisms and manydecay channels can be probed, are to a large extent due to the particular values of the Yukawacouplings. The quark and lepton couplings, and Y t in particular, are important for higgs decays,as well as to determine the production cross sections from gg fusion, higgs-strahlung, t ¯ t and W Z fusion. The LHC has (almost) measured the h τ + τ − coupling, and will also determine h µ + µ − and hb ¯ b , if they are near their SM values. Should the LHC or another future collider detect deviationsfrom the SM branching ratios or observe flavor-non-diagonal higgs decays, that would of coursebe incredibly significant (for a recent discussion, see, e.g., Ref. [71]).14 lavor Constraints on New Physics Zoltan Ligeti
Any new particle that couples to the quarks and/or leptons, potentially introduces new fla-vor violating parameters. For example, in low energy supersymmetry, which is the favorite NPscenario of a large part of our community, squark and slepton couplings may yield measurableeffects in FCNC processes and CP violation, give rise to detectable charged lepton flavor violation(CLFV), such as µ → e γ , etc. Observable CP violation is then also possible in neutral currentsand electric dipole moments, for which the SM predictions are below the near future experimen-tal sensitivities. The supersymmetric flavor problems, that TeV-scale SUSY models with genericparameters are excluded by FCNC and CP violation measurements, can be alleviated in severalscenarios: (i) universal squark masses, when ∆ ˜ m Q , ˜ D (cid:28) ˜ m (e.g., gauge mediation); (ii) alignment,when ( K dL , R ) (cid:28) m (cid:29) u L , R , ˜ d L , R , ˜ s L , R , ˜ c L , R , are all degenerate, which increases signal cross sections.Relaxing this assumption consistent with flavor bounds, results in substantially weaker squark masslimits from the LHC Run 1, around the 500 GeV scale [55]. Thus, there is a tight interplay betweenthe flavor physics and LHC high- p T searches for new physics. If there is new physics at the TeVscale, its flavor structure must be highly non-generic to satisfy current bounds, and measuringsmall deviations from the SM in the flavor sector would give a lot of information complementaryto ATLAS & CMS. The higher the scale of new physics, the less severe the flavor constraints are. IfNP is beyond the reach of the LHC, flavor physics experiments may still observe robust deviationsfrom the SM, which would point to an upper bound on the next scale to probe.
4. Final comments and ultimate sensitivity
The main points I tried to convey through some examples were: • CP violation and FCNCs are sensitive probes of short-distance physics in the SM and for NP; • Flavor physics probes energy scales (cid:29) • For most FCNC processes NP / SM > ∼
20% is still allowed, so there is plenty of room for NP; • Of the several tensions between data and SM predictions, some may soon become definitive; • Precision tests of SM will improve by 10 – 10 in many channels (including CLFV); • There are many interesting theory problems, relevant for improving experimental sensitivity; • Future data will teach us more about physics at shorter distances, whether NP is seen or not,and could point to the next energy scale to explore.With several new experiments starting (NA62, KOTO, Belle II, mu2e, COMET, etc.) and theupcoming upgrade of LHCb, the flood of new data will be fun and exciting (see Refs. [72, 73, 17]for reviews of planned flavor experiments and their sensitivities). It will allow new type of mea-surements, and more elaborate theoretical methods to be used/tested. The upcoming experiments15 lavor Constraints on New Physics
Zoltan Ligeti also challenge theory, to improve predictions and to allow more measurements to probe short dis-tance physics with robust discovery potential. Except for the few cleanest cases, improvements onboth sides are needed to fully exploit the future data sets. I am optimistic, as order of magnitudeincreases in data always triggered new theory developments, too.It is also interesting to try to estimate the largest flavor physics data sets which would be usefulto increase sensitivity to new physics, without being limited by theory uncertainties. For chargedlepton flavor violation, the SM predictions (from penguin and box diagrams with neutrinos) are(tens of) orders of magnitudes below any foreseeable experimental sensitivity, so if technologyallows significant improvements, I think the justification is obvious (as it is for electric dipole mo-ment searches). In quark flavor physics the situation is more complex. Amusingly, even in 2030,there will be theoretically clean B decay modes in which (experimental bound) (cid:14) SM (cid:38) , e.g., B → τ + τ − , B → e + e − , and probably some more. However, based on what is known today, someobservables will become limited by theory (hadronic) uncertainties. Identifying how far NP sensi-tivity can be extended is interesting, at least in principle, so below is a list for which 50/ab Belle IIand 50/fb LHCb data will not even come within an order of magnitude of the ultimately achievablesensitivities. Of course, on the relevant time scale lots of progress will take place (for estimates offuture lattice QCD uncertainties, see, Ref. [74]) and new breakthroughs are also possible. • Probably the theoretically cleanest observable in the quark sector is the determination ofthe CKM phase γ from tree-level B decays. Irreducible theory uncertainty only arises fromhigher order weak interaction [75]. So the main challenges are on the experimental side. • The theory uncertainty for the semileptonic CP asymmetries, a d , s SL , discussed is Sec. 2 and inFig 5, are also much below [76, 77] the expected 50/ab Belle II and 50/fb LHCb sensitivities. • Another set of key observables are B s , d → µ µ and B → (cid:96) ν , where the nonperturbative theoryinputs are only the decay constants, which will soon be known with <
1% uncertainties. Incontrast, the expectation for the accuracy of B d → µ µ with the full LHC data is O ( ) . • It is often stated that the determination of | V ub | is theory limited. This entirely depends on themeasurements available. In principle, the theoretically cleanest | V ub | determination I know,which only uses isospin, would be from B ( B u → (cid:96) ¯ ν ) / B ( B d → µ + µ − ) [78]. • I think that the SM prediction for CP violation in D – D mixing is below the expectedsensitivities on LHCb and Belle II. To establish this robustly, however, more theory work isneeded (especially given the recent history of hints of CP violation in D decay). • For K + → π + ν ¯ ν and especially for K L → π ν ¯ ν , the current plans for NA62 and KOTO willstop short of reaching the ultimate sensitivity to NP achievable in these decays.Thus, I guess(timate) that ∼
100 times the currently envisioned 50/ab Belle II and 50/fb LHCbdata sets would definitely allow for the sensitivity to short distance physics to improve. Whetherany of these ultimate sensitivities can be achieved at a tera- Z machine, an e + e − collider runningon the ϒ ( S ) , or utilizing more of the LHC’s or/and a future hadron collider’s full luminosity, issomething I hope we shall soon have even more compelling reasons to seriously explore. In measurements without SM backgrounds, such as setting bounds on µ → e conversion or τ → µ decay, themass-scale sensitivity (to a dimension-6 NP operator) scales like Λ ∝ ( bound ) − / . In measurements constraining SM–NP interference, Λ ∝ ( uncertainty ) − / , and at some point precise knowledge of the SM contribution becomes critical. lavor Constraints on New Physics Zoltan Ligeti
Acknowledgments
I thank Marat Freytsis, Tim Gershon, Koji Ishiwata, Michele Papucci, DeanRobinson, Josh Ruderman, Filippo Sala, Karim Trabelsi, Phill Urquijo, and Mark Wise, for recentcollaborations and/or discussions that shaped the views expressed in this talk. This work wassupported in part by the Director, Office of Science, Office of High Energy Physics of the U.S.Department of Energy under contract DE-AC02-05CH11231.
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